Adaptive demodulating method for generating replica and demodulator thereof

ABSTRACT

To prevent a coefficient vector for a modulated wave candidate of repeated identical codes from diverging, an error calculating part (36) calculates the difference between a vector X(i) with elements of input signals x(i) at N successive times and a vector Y m  (i) of N replica signals y m  (i) received from a transversal filter (34) to obtain an error vector E m  (i), the square of the norm of which is supplied to a maximum likelihood sequence estimating portion (31), when signal estimation is carried out. In addition, a code sequence candidate {a m  } is produced and a corresponding modulated wave candidate s m  is generated. The filter (34) calculates the inner product of the modulated wave candidate s m  and the coefficient vector W m  (i-1) corresponding to each state to produce y m  (i). W m  (i-1) of each state is updated to W m  (i) using an inner product vector of a generalized inverse matrix produced from s m  corresponding to the state transition selected by the estimating portion (31) and E m  (i).

TECHNICAL FIELD

The present invention relates to a demodulating method and a demodulatorfor demodulating an input signal using a replica that is adaptivelygenerated for the transmitted signal from a transmission line which hasvarying transmission characteristics.

RELATED ART

Transmission characteristics of a communication transmission line, suchas impulse response from a transmitter to a receiver, sometimescontinuously vary largely. This frequently occurs in micro-wave radiotransmission and mobile communications. In such a transmission line,when a signal is received in a relatively high noise level, a desiredsignal may contain a waveform distortion which varies with time. Inaddition, when varying interferences of the same channel and an adjacentchannel are superimposed, the transmission performance will beremarkably impaired. Thus, the impairment due to such causes should besuppressed so as to realize a receiving system with high reliability.

To receive a transmission signal from a transmission line in whichtransmission characteristics vary, an adaptive receiver using anadaptive algorithm has been used.

In a transmission line which causes a varying distortion, adaptiveequalizers have been used as the adaptive receivers. From the view pointof the arrangements of the adaptive equalizers, the adaptive equalizerscan be classified into linear equalizers and non-linear equalizers.

First, a linear equalizer will be described. FIG. 1 is a block diagramshowing the configuration of a conventional linear equalizer. In thefollowing description, a modulated received signal is represented in acomplex notation. In the complex notation, a real part of an inputsignal x(i) to the equalizer at a time point i in discrete time atintervals of one symbol represents the amplitude of the in-phasecomponent of the received signal. On the other hand, the imaginary partrepresents the amplitude of the quadrature component of the receivedsignal. The input signal x(i) is supplied from an input terminal 11 to atransversal filter 12 with M taps. By controlling tap coefficients w₁(i), . . . , w_(M) (i), the distortion of the input signal x(i) isremoved and the resultant signal is sent to a decision device 13. Thedecision device 13 outputs a decided signal d(i) from an output terminal12. The input signal and the output signal of the decision device 13 aresupplied to an error calculating portion 15. The error calculatingportion 15 calculates an error signal e(i). The error signal is sent toa control portion 16. The control portion 16 updates the tapcoefficients of the transversal filter 12 based on the error signal e(i)and the input signal x(i). The operation of the linear equalizer isdescribed in, for example, J. G. Proakis, "Digital Communications," 2ndedition, McGraw-Hill, 1989.

A column vector of M tap coefficients w₁ (i), . . . , w_(M) (i) whichare supplied to the transversal filter 12 with M taps is denoted by acoefficient vector W(i). A column vector of M input signals x(i), . . ., x(i-M+1) from a time point i to a past time point (i-M+1)corresponding to the respective tap positions is denoted by an inputvector Z(i). The input signal x(i), which is an element of the vectorZ(i), is a superimposed signal of a directly received wave signal, adelayed received wave signal, an interfering received wave signal, andnoise. Over the radio transmission line, the input signal x(i)continuously varies. A coefficient vector W(i-1) is successively updatedto W(i) based on the input vector Z(i) and the error signal e(i).

For example, when the input vector Z(i), which is the M input signalsfrom the time point i to the time point (i-M+1), is applied to thetransversal filter 12 at the time point i, the output s(i) of thetransversal filter 12 can be given by the following linear expression:

    w.sub.1 *(i)×(i)+W.sub.2 *(i)×(i-1)+. . . +w.sub.M *(i)×(i-M+1) =s(i)                                  (01)

where * represents a complex conjugate. Having obtained a plurality ofsets of measured values x(i), . . . , x(i-M+1) each set is substitutedinto Expression (01). An error e(i) between each of the resultant s(i)of the plurality of expressions and the output of the decision device 13is obtained and the coefficients w₁ (i), . . . , w_(M) (i) of Expression(01) are determined by the least square method so that the sum of thesquares of the absolute values of the errors e(i) becomes minimum.

Using an input vector Z^(H) (i)=(x*(i), . . . , x*(i-M+1)) and acoefficient vector W^(H) (i)=(w₁ *(i), . . . , w_(M) *(i)), Expression(01) is represented by the following expression:

    W.sup.H (i)Z(i)=s(i)                                       (02)

where ^(H) represents a complex conjugate transposition. When K sets ofinput signals x(i-k), x(i-1-k), . . . , x(i-M+1-k), where k=0, 1, . . ., K-1, that are sets of measured values are substituted into Expression(02), K linear expressions can be expressed by a matrix form W^(H)(i)X^(H) (i)=S^(H) (i) namely, X(i)W(i) =S(i), where X^(H) (i)=(Z(i),Z(i-1), . . . , Z(i-K+1); S^(H) (i)=(s(i), . . . , s(i-K+1); and K is aninteger equal to or greater than one. Forming a column vector by anarray of M decision signals d(i) as a vector D^(H) (i)=(d(i), d(i-1), .. . , d(i-K+1)) and further forming a column vector E(i)=D(i)-S(i), thenorm J(i)=E ^(H) (i)E(i) can be defined. Using J(i) as a cost function,the following equation is obtained from a partial differential equation∂ J(i)/∂W*(i)=0:

    X.sup.H (i)X(i)W(i)=X.sup.H (i)D(i)                        (03)

where X^(H) (i)X(i) represents an auto-correlation matrix of the inputsignal; and X^(H) (i)D(i) represents a cross-correlation vector betweenthe input signal and the decision signal. Denoting X^(H) (i)X(i) andX^(H) (i)D(i) by R(i) and V(i), respectively, Expression (03) becomesthe following expression:

    R(i)W(i)=V(i)                                              (04)

Thus, in the system shown in FIG. 1, W(i) is a solution according to themethod of least squares. The auto-correlation matrix R(i) obtained fromthe input vector Z(i) with M elements is an M-dimensional square matrix.The cross-correlation vector V(i) between the input vector Z(i) and thedecision output signal vector D(i) is an M-dimensional vector. Using thematrix R(i) and the vector V(i), the coefficient vector is representedby W(i)=R⁻¹ (i)V(i). In other words, the coefficient vector W(i) is asolution of the normal equation R(i)W(i)=V(i) and can be solved by themethod of least squares. When an inverse matrix of R(i) exists, R(i) isreferred to as a regular matrix.

However, the inverse matrix R⁻¹ (i) of the auto-correlation matrix R(i)does not always exist. Thus, there has been a problem that when such awave having an auto-correlation matrix with a rank less than M isreceived, for example, a received wave modulated with a succession ofidentical codes is received in a low noise condition, all the elementsof R(i) result in the same value, therefore, the inverse matrix R⁻¹ (i)diverges and thereby the coefficient vector W(i) diverges.

Adaptive algorithms that recursively obtain solutions of the equationsare well-known. Examples of such adaptive algorithms are Kalman filter,RLB, and LMB. For details of such adaptive algorithms, refer to Haykin,"Adaptive Filter Theory", Prentice-Hall, 1991. In these methods, whenthe auto-correlation matrix is not regular, or singular, the solutiondiverges.

As a method for obtaining a solution that does not diverge even when theauto-correlation matrix R(i) is singular, a technique using ageneralized inverse matrix is known. For the generalized inverse matrix,refer to A. Albert, "Regression and the Moore-Penrose Pseudoinverse",Academic Press, 1972. Using the generalized inverse matrix, the solutioncan be prevented from diverging even when the auto-correlation matrixR(i) is not regular. However, the solution W(i) becomes a minimum normsolution in which the norm ∥W(i)∥ of the tap coefficient vector becomesminimum and as the time i passes, since the solution of the minimum normis not identical at each time point, the solution does not alwaysconverge gradually to the true solution.

When the auto-correlation matrix R(i) is singular, an orthogonalprojection method using the Moore-Penrose generalized inverse matrix isknown as a method for approaching to the true solution asymptotically.This method is described in K. Ozeki and T. Umeda, "An adaptivefiltering algorithm using an orthogonal projection to an Affine subspaceand its properties," Trans. IECE of Japan, vol. J67-A, no. 2, pp.126-132, February 1982. However, in a transmission line in whichtransmission characteristics vary, since the input vector Z(i) changeswith time, the auto-correlation matrix R(i) obtained from X(i) alsovaries, and therefore the Moore-Penrose generalized inverse matrixshould be successively updated. Since this update processing requires alarge amount of calculation, this method is difficult to use forrealtime processing.

Although adaptivity of the linear equalizers have been improved asdescribed above, there is still a problem that sufficient equalizingeffects cannot be obtained under such a distortion condition with thenon-minimum phase in which the level of the delayed wave is larger thanthat of the direct wave.

On the other hand, non-linear equalizer configurations have been studiedin order to obtain enough equalizing effects even under a distortioncondition with the non-minimum phase. FIG. 2 shows a configuration of ademodulator that is constructed as a non-linear equalizer. An inputsignal x(i) is supplied from an input terminal 11 to a subtractor 17, inwhich a difference between the input signal x(i) and a replica signaly_(m) (i) supplied from a transversal filter 18 is calculated to obtainan error signal e_(m) (i). A squarer 19 calculates the square of theabsolute value of the error signal e_(m) (i), and the calculated resultis sent to a maximum likelihood sequence estimating circuit 21 toestimate a signal. The maximum likelihood sequence estimating circuit 21outputs a code sequence candidate {a_(m) } where m represents a statetransition candidate number. A modulating circuit 22 modulates the codesequence candidate {a_(m) } in the same manner as that in thetransmitter and outputs a modulated wave candidate s_(m), for example, acomplex symbol candidate. The modulated wave candidate s_(m) is sent tothe transversal filter 18 with M taps, where a replica signal y_(m) (i)of a desired signal is produced.

The state at the time point i is defined by a sequence of M-1 complexsymbol candidates s(i-1), s(i-2), . . . , s(i-M+1). Each complex symbolcandidate has a transition state corresponding to a code sequencecandidate {a_(m) }.

The candidates {a_(m) } from the maximum likelihood sequence estimatingcircuit 21 exist in correspondence to all state transitions at eachstate. Thus, for the same input signal x(i) the maximum likelihoodsequence estimating circuit 21 performs the above described calculationfor all candidates whose number is the product of the number of statesand the number of state transitions. The maximum likelihood sequenceestimating circuit 21 selects a most likely one of a plurality of statetransitions merging into the present time point from the previous timepoint based on the associated error signals. The control circuit 23updates the tap coefficients of the transversal filter 18 using both thecorresponding error signal e_(m) (i) and the modulated wave candidates_(m) alone the most likelihood state transition selected for each stateat the present time point. The operation of the non-linear equalizer isdescribed in, for example, K. Fukawa and H. Suzuki, "Adaptiveequalization with RLS-MLSE for frequency-selective fast fading mobileradio channels," IEEE Globecom '91, pp. 16.6.1-16.6.5, December 1991 andH. Yoshino, K. Fukawa, and H. Suzuki, "Adaptive equalization withRLS-MLSE for fast fading mobile radio channels," IEEE Inter, Symp.circuit and Sys., pp. 501-504, San Diego, May, 1992. In this manner, inthe non-linear equalizer shown in FIG. 2, the characteristics of thetransversal filter 18, or the tap coefficients w₁ (i), . . . , w_(M)(i), are controlled so as to simulate the transmission characteristicsof the transmission line.

Let W_(m) (i) represent a coefficient vector of tap coefficients fromw_(m).1 (i) to w_(m) M (i) of the transversal filter 18 with M taps forthe modulated wave candidate s_(m) (i), and S_(m) (i) be a modulatedwave candidate vector of modulated wave candidates from s_(m) (i) tos_(m) (i-M+1) over past M-1 duration. A coefficient vector W_(m) (i-1)is updated to W_(m) (i) by using a modulated wave candidate vector s_(m)(i) and the error signal e_(m) (i). When the system ideally operates,the coefficient vector W_(m) (i) is the solution of the method of leastsquares. The coefficient vector W_(m) (i) can be represented by W_(m)(i)=R_(m) ⁻¹ (i)V_(m) (i) using an auto-correlation matrix R_(m) (i)obtained from the modulated wave candidate vector S_(m) (i) and across-correlation vector V_(m) (i) between the modulated wave candidatevector S_(m) (i) and the input signal x(i). However, the inverse matrixR_(m) ⁻¹ (i) of the auto-correlation matrix has been given a conditionthat R_(m) (i) should be regular. Thus, when a modulated wave candidatehas the same successive codes, the rank of the auto-correlation matrixof the modulated wave candidate vector S_(m) (i) becomes lower than M,causing a problem that the coefficient vector W_(m) (i) diverges.

As one type of non-linear equalizer, a blind Viterbi equalizer has beenknown which does not require a training signal. The operation of thisequalizer is described in, for example, Y. Furuya, A. Ushirokawa, H.Isa, and Y. Satoh, "A study of blind Viterbi equalization algorithm",1991 Spring Nat. Convo of IEICE, A-141, March 1991. Even in thiseqalizer, for a modulated wave candidate with the same successive codes,the coefficient vector diverges. Therefore, in this equalization, when amodulated wave candidate with the same successive codes is selected, thecoefficient vector is not updated.

A method for using a generalized inverse matrix for the non-linearequalizer is described in Y. Sato, "Blind equalization and blindsequence estimation," IEICE Trans. Commun., vol. E77-B, no. 5, pp.525-556, May 1994. As described in this reference, in the conventionalmethod, all solutions take the minimum norm solution. As will bedescribed later, there is a problem that the minimum norm solutiondepends on the data sequence being transmitted, and the solution doesnot always converge to the single solution.

Other than the adaptive equalizer there are various adaptive receivers,such as interference cancelers or diversity arrangement of equalizersand cancelers. These adaptive receivers can be also classified into thelinear and non-linear types as described in (1) H. Yoshino and H.Suzuki, "Interference cancelling characteristics of DFE transversalcombining diversity in mobile radio environment--comparisons with metriccombining schemes--," Trans. IEICE of Japan vol. 76-B-II, no. 7, pp.582-595, July 1993, (2) H. Yoshino, K. Suzuki and H. Suzuki,"Interference cancelling equalizer (ICE) for mobile radiocommunications", IEEE Inter. Conf. Commun., pp. 1427-1432, May 1994, and(3) K. Fukawa and H. Suzuki, "Blind interference cancelling equalizerfor mobile radio communications", IEICE Trans. commun., vol. E77-B, no.5, pp. 580-588, May 1992. However, these adaptive receivers have thesame drawbacks as those described above.

An object of the present invention is to provide an adaptivedemodulating method and a demodulator thereof for generating a replicacorresponding to an input signal without causing divergence of acoefficient vector even when a transversal filter is supplied with amodulated wave candidate of the same successive codes and, instead,gradually converging to a true solution.

DISCLOSURE OF THE INVENTION

The adaptive demodulator according to the present invention includes: amaximum likelihood sequence estimating means for estimating a maximumlikelihood sequence based on an error signal sequence, a replicagenerating means for calculating the inner product of a coefficientvector corresponding to each state of the maximum likelihood sequenceestimating means and a modulated wave candidate vector corresponding toa state transition of each state of said maximum likelihood sequenceestimating means to produce a replica, a difference calculating meansfor calculating the difference between a replica vector with elements ofa sequence of the generated replicas and an input signal vector withelements of a sequence of the input signals to produce an error vector,and an updating means for updating the coefficient vector by using ageneralized inverse matrix generated from a modulated wave candidatecorresponding to the signal sequence selected by the maximum likelihoodsequence estimating means using the error vector.

The adaptive demodulating method according to the present inventionincludes the steps of: (1) causing maximum likelihood sequenceestimating means to output a signal sequence candidate corresponding toa state transition for each state to replica generating means, (2)causing the replica generating means to generate a modulated wavecandidate vector from a signal sequence candidate corresponding to thestate transition and calculate the inner product of the modulated wavecandidate vector and a coefficient vector corresponding to each state ofthe maximum likelihood sequence estimating means, (3) causing errorcalculating means to generate a replica vector with elements of asequence of generated replicas and an input signal vector with elementsof a sequence of input signals and calculate the difference betweenthese vectors, to produce an error vector, (4) causing the maximumlikelihood sequence estimating means to select a most likely statetransition candidate based on the error signal sequence corresponding tothe state transition of each state, and further output a decision signalaccording to the most likely state, and (5) causing updating means tocalculate the inner product of a generalized inverse matrix generatedfrom a modulated wave candidate corresponding to the state transitionselected by the maximum likelihood sequence estimating means using theerror vector, and update the coefficient vector of each state with theinner product.

The adaptive demodulator and demodulating method according to thepresent invention are different from the conventional maximum likelihoodsequence estimating demodulator and demodulating method in the followingrespects.

(1) The conventional error calculating means calculates the differencein scalar quantity and therefore the error is also in scalar quantity.On the other hand, according to the present invention, the difference iscalculated in vector quantity and therefore the error is also in vectorquantity. In the special case where the number of vector elements isone, the error calculating means of the present invention becomesequivalent to the conventional error calculating means.

(2) The conventional maximum likelihood sequence estimating meansestimates a state by using an error in scalar quantity. On the otherhand, in the present invention, state estimation is performed using anerror in vector quantity. In the special case where the number of vectorelements is one, the state estimating means of the present inventionbecomes equivalent to the conventional state estimating means.

(3) The conventional updating means obtains a coefficient vector of eachstate as the minimum norm solution of the normal equation or as aconverging solution of a recursive updating expression like a Kalmanfilter, RLS, or LMS. Unlike the conventional updating means, theupdating means of the present invention updates the coefficient vectorfor each state by using a generalized inverse matrix.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a block diagram showing a demodulator constructed as aconventional linear equalizer;

FIG. 2 is a block diagram showing a demodulator constructed as aconventional non-linear equalizer;

FIG. 3 is a block diagram showing a configuration of an adaptivedemodulator of the present invention in the case of using a maximumlikelihood sequence estimation;

FIG. 4 is a schematic diagram showing a trellis of a maximum likelihoodsequence estimating portion;

FIG. 5 is a schematic diagram for explaining update of coefficientvector;

FIG. 6 is a block diagram showing a configuration of an adaptivedemodulator according to the present invention in the case of usingdecision feedback;

FIG. 7 is a block diagram showing a configuration of an adaptivedemodulator as an interference canceler; and

FIG. 8 is a block diagram showing a configuration of an adaptivedeemodulator applied to a diversity receiver.

BEST MODES FOR CARRYING OUT THE INVENTION

FIG. B shows an embodiment of the present invention. An adaptivedemodulator of this embodiment is constructed as a non-linear equalizerusing a maximum likelihood sequence estimation as with the related artreference shown in FIG. 2. The adaptive demodulator shown in FIG. 3comprises a maximum likelihood sequence estimating portion 31, an errorcalculating portion 32, a squarer 37, a replica generating portion 38,and an updating portion 21. The error calculating portion 32 comprisesan input signal memory BB, a replica memory 35, and a subtractor 96. Theinput signal memory 33 is constructed of, for example, M stages of shiftregister that stores complex conjugate of M input signals x(i), x(i-1),. . . , x(i-M+1) from a present time point i to a past time point(i-M+1) as a column vector X(i). The replica memory 35 is constructed ofM stages of shift registers that holds M replicas y_(m) (i), y_(m)(i-1), . . . , y_(m) (i-M+1) from the time point i to the past timepoint (i-M+1) generated by the replica generating portion 38 as a columnvector Y_(m) (i). The subtractor 36 calculates the difference betweenthe input vector X(i) and the replica vector Y_(m) (i) stored in thememories 33 and 35. An output error vector E_(m) (i) of the subtractor36 is supplied to the squarer 37, which calculates the square of theabsolute value of the output error vector E_(m) (i). The output of thesquarer 37 that is a value corresponding to the likelihood is sent tothe maximum likelihood sequence estimating portion 31.

As has been well-known, the operation of the maximum likelihood sequenceestimating portion 31 can be described by its state. For example, in thecase where the number of taps M of the transversal filter 34 is two(namely, one symbol delay), and the modulating portion 39 carries outQPBK modulation, there are four states. FIG. 2, is a trellis diagramshowing states S₁, S₂, S₃, and S₄. From each state of S₁, S₂, S₃, and S₄at a time point (i-1) there can be a transition to any one of the statesS₁, S₂, S₃, and S₄ at a time point i. At the time point (i-1), acoefficient vector W_(m) (i-1) corresponding to each state m is set tothe transversal filter 32.

An input signal x(i) is supplied from an input terminal 11 to the errorcalculating portion 32. A sequence of input signals at consecutive Ntime points are stored in the input signal memory 33. The input signalmemory 33 outputs an input signal vector X(i) with elements of thesequence of the input signals x(i) given by the following expression.

    X.sup.H (i)=(x(i), x(i-1), . . . , x(i-M+1))               (1)

As is clear from Expression (1), the definition of the input signalvector X(i) differs from that of the input vector Z(i) of the abovedescribed linear equalizer. Each replica signal y_(m) (i) output fromthe transversal filter 32 is supplied to the replica memory 35, thus, asequence of N replica signals are stored in the replica memory 35 of theerror calculating portion 32. The replica memory 35 outputs a replicavector Y_(m) (i) with elements of the sequence of the replica signalsy_(m) (i). The replica signals y_(m) (i) is given by the followingexpression:

    Y.sub.m.sup.H (i)=(y.sub.m (i), y.sub.m (i-1), . . . , y.sub.m (i-M+1))

The subtractor 36 calculates the difference between the input signalvector X(i) and the replica vector Y_(m) (i) and outputs an error vectorE_(m) (i).

The square of the norm of the error vector E_(m) (i) is calculated bythe squarer 37 and-the result is provided to the maximum likelihoodsequence estimating portion 31. The maximum likelihood sequenceestimating portion 31 estimates a signal and the estimated signal isoutput from an output terminal 14. The maximum likelihood sequenceestimating portion 31 outputs a code sequence candidate {a_(m) } to thereplica generating portion 38. The replica generating portion 38consists of a modulating portion 39 and the transversal filter 32. Themodulating portion 39 performs the same modulation with the codesequence candidate {a_(m) } as the transmitting side does and outputs amodulated wave candidate {s_(m) }. The modulated wave candidate {s_(m) }is applied to the transversal filter 32. The transversal filter 34calculates the inner product (convolution) of the modulated wavecandidate {a_(m) } and the coefficient vector W_(m) (i-1) correspondingto each state transition candidate m at the maximum likelihood sequenceestimating portion 31 to produce a replica signal y_(m) (i).

As shown in FIG. 4, to each of the states S₁ to S₄ at the time point i,there are state transitions from the respective states at the time point(i-1). Thus, four state transitions are merging. From these four statetransitions, one state transition with the maximum likelihood, or astate transition with small ∥E_(m) (i)∥², is selected. In FIG. 2,selected state transitions to each state at the time point i are denotedby thick lines. The updating portion 21 according to the presentinvention updates the coefficient vector W_(m) (i-1) for each state toW_(m) (i) by calculating an inner product vector of a generalizedinverse matrix, which is generated from the modulated wave candidate{s_(m) } corresponding to the state transition selected by the maximumlikelihood sequence estimating portion 31, and the error vector E_(m)(i). Since the maximum likelihood sequence estimating portion 31 outputscandidates {a_(m) } corresponding to the number of state transitions ateach state, the calculations must be carried out corresponding statenumber and state transition number for the single input signal x(i).

Practically, W(i) is updated as follows. First, the principle of theupdating algorithm will be described. A modulated wave candidate vector(complex symbol candidate vector)

    S.sub.m.sup.H (i)=(s.sub.m *(i), (s.sub.m *(i-1), . . . , s.sub.m *(i-N+1))

is generated corresponding to a state transition candidate mcorresponding to a code sequence candidate {a_(m) } selected by themaximum likelihood sequence estimating portion In addition, a modulatedwave candidate matrix

    A.sub.m.sup.H (i)=(S.sub.m (i), S.sub.m (i-1), . . . , S.sub.m (i-N+1))

is generated corresponding to N modulated wave candidate vectors over Nconsecutive time points, where N represents an integer that is 1 orgreater. Using A_(m) (i), an auto-correlation matrix R_(m) (i) of themodulated wave candidate and cross-correction vector V_(m) (i) betweenthe modulated wave candidate and the input signal are calculated asfollows:

    R.sub.m (i)=A.sub.m.sup.H (i)A.sub.m (i)                   (2)

    V.sub.m (i)=A.sub.m.sup.H (i)X(i)                          (3)

R_(m) (i) and V_(m) (i) are calculated for each state transitioncandidate m.

A vector W_(m) (i) is obtained by solving the normal equation expressedby R_(m) (i) and V_(m) (i) as

    R.sub.m (i)W.sub.m (i)=V.sub.m (i)

Supporting that when W_(r) is an arbitrary vector having the samedimensions as W_(m) (i), a general solution of W_(m) (i) is given by thefollowing expression.

    W.sub.m (i)=R.sub.m.sup.+ (i)V.sub.m (i)+(I.sub.M -R.sub.m.sup.+ (i)R.sub.m (i))W.sub.r                                               (4)

where I_(M) represents an M by M unit matrix. The proof of thisexpression is described in, for example, the above described referencewritten by A. Albert.

In Expression (4), the Moore-Penrose generalized inverse matrix R_(m) ⁺(i) should be calculated based on R_(m) (i). Several mathematicalmethods are known for calculating R_(m) ⁺ (i) from R_(m) (i). Forexample, in Chapter 5 of the reference written by A. Albert, fourmethods are described in detail.

The first term in the right hand side of Expression (4), which is asolution obtainable by assuming W_(r) =0, is the particular solutionequivalent to the minimum norm solution. For a state transitioncandidate m which renders R_(m) (i) regular, R_(m) ⁺ (i)=R_(m) ⁻¹ (i)holds and the second term of the right side of Expression (4) iscancelled; therefore, the minimum norm solution of the first termbecomes the solution. However, since the signal sequence is transmittedat random, R_(m) ⁺ (i) does not always become R_(m) ⁻¹ (i). In theconventional method which allows only the minimum norm solution, thereis a drawback that a true solution cannot be obtained because R_(m) ⁺(i) changes according to the random signal sequence. In such a case, byusing the general solution, W_(m) (i) can be obtained in the followingmanners:

First Updating Method

For arbitrary vector Wr with the same dimensions as W(i),Expression (4)holds. Equating Wr with W_(m) (i-1), Expression (4) becomes thefollowing recursive expressions:

    W.sub.m (i)=W.sub.m (i-1)+R.sub.m.sup.+ (i)Δ .sub.m (i) (5)

    Δ .sub.m (i)=V.sub.m (i)-R.sub.m (i)W.sub.m i-1)     (6)

Expression (6) represents a prediction error. In Expression (5), W_(m)(i-1) is corrected by using the prediction error and updated to W_(m)(i).

The cross-correlation vector V_(m) (i) of Expression (6) is calculatedby Expression (3) using the input signal vector X(i) and the modulatedwave candidate transposed matrix A_(m) ^(H) (i). In Expressions (5) and(6), the auto-correlation matrix R_(m) (i), defined by Expression (2)using the modulated wave candidate, and the generalized inverse matrixR_(m) ⁺ (i) are obtained for the state transition m, which means thatthese matrices are not based on actual received input signal. Instead,they are based upon the auto-correlation function of a pure signalcomponent free of fluctuation of transmission characteristics and noiseaddition. Thus, R_(m) (i) and R.sub.⁺ (i) are given only by theknowledge of each state, and they can be calculated beforehandcorresponding to the state transition candidate m. Then, the matricesR_(m) (i) and R_(m) ⁺ (i) calculated beforehand may be prestored in amatrix memory 41M of the updating portion 41 in correspondence with thestate transition candidate m, and each time a code sequence candidate isdesignated by the maximum likelihood sequence estimating portion 31, acorresponding state transition candidate m is provided to the updatingportion 41 to read out the corresponding auto-correlation matrix R_(m)(i) and the generalized inverse matrix R_(m) ⁺ (i) from the matrixmemory 41M and calculate the tap coefficient vector W_(m) (i) byexpressions (5) and (6). This method tremendously reduces thecalculation amount necessary for the demodulation processing.

The above described first updating algorithm holds for arbitrary N forthe dimension M. When N=1, A_(m) ^(H) (i)=S_(m) (i) and A_(m) (i)=S_(m)^(H) (i) are obtained and, therefore, R_(m) (i)=S_(m) (i)S_(m) ^(H) (i)and V_(m) (i)=S_(m) (i)X(i) are obtained. In this case, either of theinput signal memory 33 and the replica memory 35 can consist of onestage, and the hardware can be simplified.

Second Updating Method

Applying the following matrix equations,

    R.sub.m.sup.+ (i)V.sub.m (i)=A.sub.m.sup.+ (i)X(i)         (7)

    R.sub.m.sup.+ (i)R.sub.m (i)=A.sub.m.sup.+ (i)A(i)         (8)

to the expressions (5) and (6), the second updating algorithm is givenby

    W.sub.m (i)=W.sub.m (i-1)+A.sub.m.sup.H (i)Δ'.sub.m (i) (9)

    Δ'.sub.m (i)=X(i)-A.sub.m (i)W.sub.m (i-1)           (10)

Since this updating method does not require calculations of R_(m) (i)and V_(m) (i), the calculation can be simplified. The matrix A_(m) (i)and its generalized inverse matrix A_(m) ⁺ (i) calculated beforehandcorresponding to the state transition candidate m are prestored in thematrix memory 41M, and can be read out therefrom upon request, whichresults in reduction of the calculation amount. The right hand side ofExpression (10) is the same as the output E_(m) (i) of the subtractor 36shown in FIG. 3. Thus, when the output E_(m) (i) of the subtractor 36 isused as denoted by a broken line instead of calculating it by Equation(10), the calculation amount can be further reduced. In this case, sinceA_(m) (i) is not used, it suffices only to prestore A_(m) ⁺ (i) in thematrix memory 41M.

Third Updating Method

The third updating algorithm introduces a step size μ into Expression(5) of the first updating method as follows:

    W.sub.m (i)=W.sub.m (i-1)+μR.sub.m (i)Δ.sub.m (i) (11)

The value of the step size μ is positive and around 1. When the value ofμ increases, the coefficient vector W_(m) (i) can quickly converge;however, the convergence error becomes large. On the contrary, when thevalue of μ decreases, the coefficient vector W_(m) (i) slowly converges.Thus, depending on the application, it is possible to adjust theconvergence characteristics of the coefficient vector. In this case, thecalculation amount of the demodulation decreases if calculated R_(m) (i)and μR_(m) ⁺ (i) are calculated beforehand and prestored in the matrixmemory 41M. Similarly, for Expression (9) of the second updating method,the step size μ can be introduced as follows:

    W.sub.m (i)=W.sub.m (i-1)+μA.sub.m.sup.+ (i)Δ'.sub.m (i) (12)

In the above described updating algorithms, the auto-correlation matrixR_(m) (i) and the cross-correlation vector V_(m) (i) have been definedas R_(m) (i)=A_(m) ^(H) (i)A_(m) (i) and V_(m) (i)=A_(m) ^(H) (i)X(i),respectively. Whereas they can be exponentially weighted by defining asR_(m) (i)=A_(m) ^(H) (i)Λ A_(m) (i) and V_(m) (i)=A_(m) ^(H) (i)Λ X_(m)(i), where A represents a diagonal matrix Λ=diag(1,λ, . . . ,λ^(N-1)); λrepresents a constant forgetting factor satisfying 0<λ<1. In such acase, the coefficient vector can be updated by Expression (4), or by thefirst updating method according to Expressions (5) and (6). In thesecond updating method, however Expressions (7) and (8) hold only whenthe rank of A_(m) (i) is N and N≦M, which cannot hold, generally.Consequently, only when Expressions (7) and (8) hold, the coefficientvector can be updated by Expressions (9) and (10).

The vector W_(m) (i) updated by the above expression is not the minimumnorm solution, i.e., not a solution which makes ∥W_(m) (i)∥ minimum, buta solution which makes the difference norm ∥W_(m) (i)-W_(m) (i-1)∥ orequivalently ∥W_(m) (i)-W_(r) ∥ minimum. This means that, in theupdating algorithm with the true solution denoted by W(i), a straightline that connects the vector W_(m) (i-1) and the vector W_(m) (i) inthe vector space becomes perpendicular to the straight line thatconnects the vector W_(m) (i) and the vector W(i) as shown in FIG. 5.Therefore, let the distance from W(i) to W_(m) (i), and the distancefrom W(i) to W_(m) (i-1) be represented by L₁ =∥W(i)-W_(m) (i)∥ and L₂=∥W(i)-W_(m) (i-1)∥, respectively, the relation L₁ ≦L₂ can be derived byapplying the Pythagorean theorem. Therefore, while a randomly modulatedsignal is being received, as the time i passes, the coefficient vectoralways approaches the true solution. This characteristic is effectivelyapplicable to blind type equalizers and the like.

Since a code sequence corresponding to a state transition m for eachstate is usually fixed, the modulated wave candidate vector S_(m) (i)for each state transition is fixed. In other words, the modulated wavecandidate matrix A_(m) (i) can be regarded as a constant matrix A_(m)that does not vary with time. Thus, the updating portion 41 cancalculate the generalized inverse matrix R_(m) ⁺ (i) or A_(m) ⁺beforehand and prestores it in the matrix memory 41M of the updatingportion 41. In the actual applications, since the matrix prestored inthe matrix memory 41M can be used for the calculation, the calculationamount and the processing time can be reduced. This property isremarkably different from the above described orthogonal projectionmethod for the linear equalizer in which a received signal containingnoises is fed into the transversal filter. Although the expressions canbe modified in various manners associated with the above described threealgorithms, those matrices which can be calculated from A_(m) (i), likeR_(m) ⁺,R_(m), and so forth may be calculated beforehand and prestoredin the matrix memory 41M.

In the non-linear equalizer, as the delay time increases, the number ofstates and processing amount exponentially increase. To solve such aproblem, various state reducing methods have been studied as in A.Duel-Hallen, C. Heegard, "Delayed decision feedback sequenceestimation," IEEE Trans. Communi., vol. 38, no. 5, pp. 228-236, May1986. Like this reference, the maximum likelihood sequence estimationwith reduced state number refers to a code sequence stored in a pathmemory to define effective states. Thus, it is necessary to calculate,beforehand, a generalized inverse matrix having incorporated therein thecode sequence stored in the path memory. This concept can be applied toan adaptive demodulator constructed by a decision feedback equalizer(DFE) shown in FIG. 6.

In the embodiment shown in FIG. 6, the transversal filter 32 generates adelayed wave component of a multi-path transmission line. A subtractor36 removes the delayed wave component from the input signal x(i) that issupplied through a feed forward filter 43. A decision device 13 decideson a desired signal component and outputs the decision result as a codes(i). An error calculating portion 15 calculates an error e(i) betweeninput and output of the decision device 13. The output of portion 15 issquared in squarer 37, and the result is sent to an updating portion 21adopting the present invention. The updating portion 41 determines a tapcoefficient vector W(i) of the transversal filter 34 so that the valueof |e|² becomes minimum. In this embodiment, the feed forward filter 43is provided on the input side. The updating portion 41 controls acoefficient H(i) of the filter 43 so that the level of the directlyreceived wave component (desired signal component) of the input signalbecomes larger than the level of the delayed wave component.

In the embodiment shown in FIG. 6, a signal s(i) for the decisionfeedback that is the output of the decision device 13 is fed back to thesubtractor 36 through the transversal filter 34. The subtractor 36subtracts the delayed wave component from the input signal x(i) andoutputs a desired signal component. The decision device 13 decides thedesired signal component and outputs the decision result as code s(i)=d(i) which constitutes code sequence. When the signal sequence sent tothe transversal filter 34 with M taps is denoted by s_(m) (i), s_(m)(i-1), . . . , s_(m) (i-M+1) and the output of the feed forward filter43 is denoted by x(i), the modulated wave candidate vector S(i), theauto-correlation matrix R(i), and the cross-correlation vector V(i) canbe defined in the same manner. Here the single state case, m=1, isconsidered. Thus, the coefficient vector W(i) of the transversal filter32 can be updated according to the updating algorithm using Expressions(2) through (11).

It should be noted that the present invention can be applied to a blindtype non-linear equalizer, non-linear interference canceler, andreceivers with diversity construction thereof as well as the abovedescribed non-linear equalizer.

FIG. 7 is a block diagram showing a configuration of a non-linearinterference canceler according to the demodulator of the presentinvention. FIG. 8 is a block diagram showing a configuration of acombination of a non-linear equalizer and a diversity receiver.

The non-linear interference canceler shown in FIG. 7 consists oftransversal filters 34₁ and 34₂, subtractors 36₁ and 36₂, and modulatingportions 39₁ and 39₂ corresponding to a desired signal and aninterference signal, respectively. Since N=1, the memories 33 and 35shown in FIG. 3 are not provided. The maximum likelihood sequenceestimating portion 31 successively generates sets of desired signalsequence candidates and interference signal sequence candidates andsends them to the modulating portions 39₁ and 39₂, respectively. Themodulating portions 39₁ and 39₂ generate desired modulated signalsequence candidates and interference modulated signal sequencecandidates and send them to the transversal filters 34₁ and 34₂,respectively. The updating portion 41 according to the present inventioncontrols the tap coefficients of the transversal filters 34₁ and 34₂ soas to simulate the transmission characteristics of the desired signaltransmission line and the interference signal transmission line. Thetransversal filters 34₁ and 34₂ output replicas of the desired signaland the interference signal to the subtractors 36₁ and 36₂,respectively. The subtractors 36₁ and 36₂ successively subtract thereplicas from the input signals x(i). The finally subtracted result isdefined as an error vector E_(m) (i). The error vector E_(m) (i) issupplied to the squarer 37. In the embodiment shown in FIG. 7, theestimated interference signal is canceled as an unnecessary signalcomponent of the received signal x(i) so as to reduce the decision errorof the desired signal.

The demodulating method given by the present invention is applicable tothe diversity receiver shown in FIG. 8, where input signals x₁ (i) andx₂ (i) supplied to input terminals 11₁ and 11₂ from two branches,subtractors 36₁ and 36₂, squarers 37₁ and 37₂, transversal filters 34₁and 34₂, and updating portions 41₁ and 41₂ are provided, respectively.Outputs of the squarers 37₁ and 37₂ are supplied to the adder 42. Theadded result is supplied to the maximum likelihood sequence estimatingportion 31. In this embodiment, the condition of N=1 is assumed. Themodulating portion 39 modulates each signal sequence candidate. Themodulated signal sequence candidate is supplied commonly to thetransversal filters 34₁ and 34₂ and the updating portions 41₁ and 41₂corresponding to the two branches. The transversal filters 34₁ and 34₂simulate the transmission characteristics of the received signals x₁ (i)and x₂ (i) received from two different transmission lines through thetwo branches and output replicas y_(1m) (i) and y_(2m) (i),respectively. The subtractors 36₁ and 36₂ output errors E_(1m) (i) andE_(2m) (i) between the received signals x₁ (i) and x₂ (i) and thereplicas y_(1m) (i) and y_(2m) (i), respectively. The errors E_(1m) (i)and E_(2m) (i) are supplied to the squarers 37₁ and 37₂, respectively.These operations are described in the above mentioned references. As isclear from the drawing, each portion of this construction operates inthe same manner as the above described non linear equalizer. Thus, thepresent invention can be easily applied to the configuration shown inFIG. 8. In each of the above described embodiments, when the number oftaps M is 1, the demodulator is rendered incapable of having anequalizing function or cancelling function.

As described above, according to the present invention, an optimumcoefficient vector for each signal candidate is set. Each coefficientvector is successively updated by using the Moore-Penrose generalizedinverse matrix. Thus, the signal candidate can be stably obtained. Inaddition, since the Moore-Penrose generalized inverse matrix depends oneach state transition candidate, it can be calculated beforehand andprestored in the memory. Thus, the calculation amount of the presentinvention is lower than that of the conventional adaptive signalprocessing that calculates an inverse matrix. In addition, since themaximum likelihood sequence estimating portion is provided, the signaldetecting performance can be improved.

Accordingly, the receiver according to the present invention is suitablefor mobile radio communications and mobile satellite communications inwhich fading varies at high speed. In addition, the receiver accordingto the present invention is suitable for micro-wave terrestrial radiocommunications using a high precision demodulating circuit fortransmitting large amount of data. According to the present invention,adaptive receivers using adaptive equalizers, adaptive interferencecancelers, and diversity construction can be easily obtained.

I claim:
 1. An adaptive demodulator, comprising:maximum likelihoodsequence estimating means for generating signal sequence candidatescorresponding to respective state transitions at each state for an inputsignal at each of discrete time points at predetermined intervals,selecting a signal sequence candidate with a high likelihood statetransition from all state transitions by using an error vector, andoutputting the selected signal sequence candidate; replica generatingmeans for generating a modulated wave candidate vector from each saidsignal sequence candidate supplied from said maximum likelihood sequenceestimating means, and calculating the inner product of the modulatedwave candidate vector and a coefficient vector corresponding to eachstate of said maximum likelihood sequence estimating means to generate areplica; error calculating means for calculating the difference betweena replica vector with elements of a sequence of the generated replicasand an input signal vector with elements of a sequence of the inputsignals; and updating means for updating the coefficient vector at thepresent time point i by using a generalized inverse matrix produced froma modulated wave candidate corresponding to the selected signal sequencecandidate and a coefficient vector at a past time point (i-1).
 2. Theadaptive demodulator as set forth in claim 1, further comprising:inputsignal memory means for holding input signals x(i), . . . , x(i-M+1)from the time point i to a past time point (i-M+1) and outputting theinput signals as an input signal vector X(i), where x(i) represents theinput signal at the time point i and M represents an integer equal to orgreater than 2; and replica memory means for holding M replicas y_(m)(i), . . . , y_(m) (i-M+1) from the time point i to the past time point(i-M+1) supplied from said replica generating means and outputting thereplicas as a replica vector where m represents a state transition andy_(m) (i) represents the replica corresponding to the state transitionm.
 3. The adaptive demodulator as set forth in claim 2,wherein saidreplica generating means includes a transversal filter with M tapscorresponding to M coefficients of a coefficient vector W_(m) (i)corresponding to the transition states, said transversal filter beingsupplied with M elements of the modulated wave candidate vector andoutputting an inner product of the M elements and the M coefficients assaid replica, and wherein said updating means calculates a coefficientvector W_(m) (i) based on the following expressions

    W.sub.m (i)=W.sub.m (i-1)+R.sub.m.sup.+ (i)Δ.sub.m (i)

    Δ.sub.m (i)=V.sub.m (i)-R.sub.m (i)W.sub.m (i-1)

    R.sub.m (i)=A.sub.m.sup.H (i)A.sub.m (i)

    V.sub.m (i)=A.sub.m.sup.H (i)X(i)

where R_(m) (i) represents an auto-correlation matrix of the modulatedwave candidate vector, R_(m) ⁺ (i) represents a generalized inversematrix of R_(m) (i), A_(m) (i) represents a modulated wave candidatematrix with elements of the modulated wave candidate vectors S(i), . . ., S(i-N+1) from the time point i to a past time point (i-N+1), N beingan integer equal to or greater than 1, A_(m) (i) represents a complextransposed matrix of A_(m) (i) and V_(m) (i) represents a mutualcorrelation vector of the modulated wave candidate vector and the inputsignal vector.
 4. The adaptive demodulator as set forth in claim 3,wherein said updating means includes matrix memory means for storing thematrices R_(m) (i) and R_(m) ⁺ (i) that are pre-calculated correspondingto the state transition m.
 5. The adaptive demodulator as set forth inclaim 3 or 4, wherein μR_(m) ⁺ (i) is used instead of the generalizedinverse matrix R_(m) ⁺ (i), where μ represents a desired positiveconstant.
 6. The adaptive demodulator as set forth in claim 2,whereinsaid replica generating means includes a transversal filter with M tapscorresponding to M coefficients of a coefficient vector W_(m) (i)corresponding to the transition states, the transversal filter beingsupplied with M elements of the modulated wave candidate vector andoutputting an inner product of the M elements and the M coefficients assaid replica, and wherein said updating means calculates a coefficientvector W_(m) (i) based on the following expressions

    W.sub.m (i)=W.sub.m (i-1)+A.sub.m.sup.+ (i)Δ'.sub.m (i)

    Δ'.sub.m (i)=X(i)-A.sub.m (i)W.sub.m (i-1)

where A_(m) (i) represents a modulated wave candidate matrix withelements of the N modulated wave candidate vectors S(i), . . . ,S(i-N+1) from the time point i to a past time point (i-N+1), and A_(m) ⁺(i) represents a generalized inverse matrix of A_(m) (i).
 7. Theadaptive demodulator as set forth in claim 6, wherein the error vectorthat is output from said error calculating means is used for the vectora Δ'_(m) (i).
 8. The adaptive demodulator as set forth in claim 6 or 7,wherein said updating means includes a matrix memory means for storingthe matrix A_(m) ⁺ (i) that is pre-calculated corresponding to the statetransition m.
 9. The adaptive demodulator as set forth in claim 6 or 7,wherein μA_(m) ⁺ (i) is used instead of the matrix A_(m) ⁺ (i), where μrepresents a desired positive value.
 10. The adaptive demodulator as setforth in claim wherein the matrix R_(m) (i) and the vector V_(m) (i) aredefined as followsR_(m) (i)=A_(m) ^(H) (i)Λ A_(m) (i) V_(m) (i)=A_(m)^(H) (i)Λ X(i) Λ=diag (1λ. . . λ^(N-1))where N represents an integerequal to or greater than 1 and λ represents a constant satisfying 0<λ<1.11. An adaptive demodulator, comprising:a transversal filter controlledby a coefficient vector to simulate a delayed wave component of an inputsignal, for producing a simulation delayed wave component; differentialmeans for subtracting an output of said transversal filter from theinput signal and outputting a desired signal component; deciding meansfor deciding the desired signal component supplied from saiddifferential means and outputting a desired signal; and updating meanssupplied with a sequence of M desired signals from a time point i to apast time point (i-M+1) and the input signal, for calculating thecoefficient vector at the time point i by using a generalized inversematrix produced from the desired signal sequence and the coefficientvector at a time point (i-1), and supplying the coefficient vector tosaid transversal filter.
 12. An adaptive demodulator, comprising:maximumlikelihood sequence estimating means for generating desired signalsequence candidates and interference signal sequence candidatescorresponding to respective state transitions at each state for an inputsignal at each of discrete time points at predetermined intervals,selecting a desired signal sequence candidate with a high likelihoodstate transition and an interference signal sequence candidate with ahigh likelihood state transition from all state transitions by using anerror vector, and outputting the selected desired signal sequencecandidate and interference signal sequence candidate; replica generatingmeans for generating a desired modulated wave candidate vector and aninterference modulated wave candidate vector from each said desiredsignal sequence candidate and said interfered signal sequence candidatesupplied from said maximum likelihood sequence estimating means, andcalculating an inner product of the desired wave candidate vector andthe interference wave coefficient vector corresponding to each state ofsaid maximum likelihood sequence estimating means to generate a desiredwave replica and an interference wave replica; error calculating meansfor subtracting a desired wave replica vector with elements of asequence of the generated desired wave replicas and an interference wavereplica vector with elements of a sequence of the generated interferencewave replicas from an input signal vector with elements of a sequence ofthe input signals to produce the error vector; and updating means forupdating the desired wave coefficient vector at the present time point iby using a generalized inverse matrix produced from a desired modulatedwave candidate corresponding to the selected desired signal sequencecandidate and a desired wave coefficient vector at a past time point(i-1) and updating the interference wave coefficient vector at thepresent time point i by using a generalized inverse matrix produced froman interference modulated wave candidate corresponding to the selectedinterference signal sequence candidate and an interference wavecoefficient vector at the past time point (i-1).
 13. An adaptivedemodulator, comprising;maximum likelihood sequence estimating means forgenerating signal sequence candidates corresponding to respective statetransitions at each state for first and second input signals suppliedfrom first and second branches at each of discrete time points atpredetermined intervals, selecting signal sequence candidates with highlikelihood state transitions from all state transitions, and outputtingthe selected signal sequence candidates; first and second replicagenerating means for generating modulated wave candidate vectors fromeach signal sequence candidate received from said maximum likelihoodsequence estimating means, and calculating an inner product of themodulated wave candidate vectors and first and second coefficientvectors corresponding to each state of said maximum likelihood sequenceestimating means to generate first and second replicas; first errorcalculating means for calculating the difference between a first replicavector with elements of a sequence of the first replicas and a firstinput signal vector with elements of a sequence of the first inputsignals to generate a first error vector; second error calculating meansfor calculating the difference between a second replica vector withelements of a sequence of the second replicas and a second input signalvector with elements of a sequence of the second input signals togenerate a second error vector; likelihood generating means forcalculating the sum of norms of the first and second error vectors as asignal corresponding to the likelihood and supplying the signal to saidmaximum likelihood sequence estimating means; and first and secondupdating means for updating the first and second coefficient vectors atthe present time point i by using a generalized inverse matrix producedfrom a modulated wave candidate corresponding to the selected signalsequence candidate and the first and second coefficient vectors at thepast time point (i-1).
 14. An adaptive demodulating method forgenerating a replica, comprising the steps of:(1) providing a signalsequence candidate corresponding to a state transition for each statefrom maximum likelihood sequence estimating means to replica generatingmeans; (2) generating by the replica generating means, a modulated wavecandidate vector from a signal sequence candidate corresponding to thestate transition and calculating an inner product of the modulated wavecandidate vector and a coefficient vector corresponding to each state ofthe maximum likelihood sequence estimating means to produce a replica;(3) generating by error calculating means, a replica vector withelements of a sequence of generated replicas and an input signal vectorwith elements of a sequence of input signals, and producing a differencebetween these vectors as an error vector; (4) causing the maximumlikelihood sequence estimating means to select a high likelihood statetransition candidate from the error signal sequences corresponding tothe state transitions for each state and output a decided signalcorresponding to the most likelihood state; and (5) causing updatingmeans to calculate the inner product of a generalized inverse matrixproduced from a modulated wave candidate corresponding to the selectedstate transition and the error vector and update the coefficient vectorwith the inner product for each state.
 15. The demodulating method asset forth in claim 12, wherein in said step (5) said updating meanscalculates a coefficient vector W m(i) based on the followingexpressions:

    W.sub.m (i)=R.sub.m.sup.+ (i)V.sub.m (i)+(I.sub.M -R.sub.m.sup.+ (i)R.sub.m (i))Wr

    R.sub.m (i)=A.sub.m.sup.H (i)A.sub.m (i)

    V.sub.m (i)=A.sub.m.sup.H (i)X(i)

    Wr≠0

where x(i) represents an input signal at a time point i, X(i) representsan input signal vector with elements of input signals x(i), . . . ,x(i-M+1) from the time i to a past time point (i-M+1), M represents aninteger equal to or greater than 1; m represents a state transition,W_(m) (i) represents a coefficient vector with elements of coefficientsw₁ (i), . . . , w_(M) (i) corresponding to the state transition m; Wrrepresents any vector with the same dimensions as W_(m) (i), R_(m) (i)represents an auto-correlation matrix of the modulated wave candidatevector, R_(m) ⁺ (i) represents a generalized inverse matrix of R_(m)(i), I_(M) represents an M×M unit matrix, A_(m) (i) represents amodulated wave candidate matrix with elements of up to N modulated wavecandidate vectors S(i), . . . , S(i-N+1) from the time point i to thepast time point (i-N+1), N being 1 or greater, A_(m) ^(H) (i) representsa complex transposed matrix of A_(m) (i), and V_(m) (i) represents amutual correlation vector of the modulated wave candidate vector and theinput signal vector.
 16. The demodulating method as set forth in claim14, wherein in said step (5) said updating means calculates thecoefficient vector W_(m) (i) based on the following expressions:

    W.sub.m (i)=W.sub.m (i-1)+R.sub.m.sup.+ (i)Δ.sub.m (i)

    Δ.sub.m (i)=V.sub.m (i)-R.sub.m (i)W.sub.m (i-1)

    R.sub.m (i)=A.sub.m.sup.H (i)A.sub.m (i)

    V.sub.m (i)=A.sub.m.sup.H (i)X(i)

where x(i) represents an input signal at the time point i, X(i)represents an input signal vector with elements of complex conjugationof input signals x(i), . . . , x(i-M+1) from the time point i to a pasttime point (i-M+1), M represents an integer equal to or greater than 1,m represents a state transition, W_(m) (i) represents a coefficientvector with elements of coefficients w₁ (i), . . . , w_(M) (i)corresponding to the state transition m, R_(m) (i) represents anauto-correlation matrix of R_(m) (i) the modulated wave candidatevector, R_(m) ⁺ (i) represents a generalized inverse matrix of, R_(m)(i)A_(m) (i) represents a modulated wave candidate matrix with elementsof the N modulated wave candidate vectors S(i), . . . , S(i-N+1) fromthe time point i to a past time point (i-N+1), N being an integer equalto or greater than 1, A_(m) ^(H) (i) represents a complex transposedmatrix of A_(m) (i), and V_(m) (i) represents a mutual correlationvector of the modulated wave candidate vector and the input signalvector.
 17. The demodulating method as set forth in claim 15 or16wherein the matrices R_(m) (i) and R_(m) ⁺ (i) are pre-calculated andprestored in matrix memory means, and wherein said step (5) is performedin response to the designation of the state transition candidate m byreading out the matrixes R_(m) (i) and R_(m) ⁺ (i) corresponding to thedesignation of the state transition candidate m from the matrix memorymeans so as to be used for calculating the coefficient vector.
 18. Thedemodulating method as set forth in claim 15, wherein in said step (5)said updating means calculates the coefficient vector W_(m) (i) based onthe following expressions:

    W.sub.m (i)=W.sub.m (i-1)+A.sub.m.sup.+ (i)Δ'.sub.m (i)

    Δ'.sub.m (i)=X.sub.m (i)-A.sub.m (i)W.sub.m (i-1)

where x(i) represents an input signal at the time point i, X(i)represents an input signal vector with elements of complex conjugationof input signals x(i), . . . , x(i-M+1) from the time point i to a pasttime point (i-M+1), M represents an integer equal to or greater than 2 mrepresents a state transition; W(i) represents a coefficient vector withelements of coefficients w₁ (i), . . . , w_(M) (i), A_(m) (i) representsa modulated wave candidate matrix with elements of the modulated wavecandidate vectors S(i), . . . , S(i-N+1) from the time point i to a pasttime point (i-N+1), and A_(m) ⁺ (i) represents a general inverse matrixof A_(m) (i).
 19. The demodulating method as set forth in claim 18,wherein the error vector that is output from the error calculating meansis used for the vector Δ'_(m) (i).
 20. The demodulating method as setforth in claim 18 or 19,wherein the matrix A_(m) ⁺ (i) pre-calculatedcorresponding to the state transition candidate m is prestored in matrixmemory means; and wherein said step (5) is performed in response to thedesignation of the state transition candidate m by reading out thematrix A_(m) ⁺ (i) corresponding to the designation of the statetransition candidate from said matrix memory means so as to be used forcalculating the coefficient vector.
 21. The demodulating method as setforth in claim 16 or 18, wherein μ A_(m) ⁺ (i) is used instead of thematrix A_(m) ⁺ (i) where μ represents a desired positive value.
 22. Thedemodulating method as set forth in claim 16, wherein the matrix R_(m)(i) and the vector V_(m) (i) are defined as follows

    R.sub.m (i)=A.sub.m.sup.H (i)Λ A.sub.m (i)

    V.sub.m (i)=A.sub.m.sup.H (i)Λ X(i)

    Λ=diag (1 λ. . . λ.sup.N-1)

where N represents an integer equal to or greater than 1 and λrepresents a constant satisfying 0<λ<1.